Momentum density expansions

Then the radial terms of this expansion can be connected to those of the momentum density expansion of Eq. (5.36) by... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The following figures show the results of the reconstructions using the described methods. Figures 4 and 6 show the momentum density anisotropy of Cu and Cuo.953Alo.047 respectively in the (110) plane. The anisotropy is obtained by neglecting the first, isotropic term of the series expansion of p(p) in cubic lattice harmonics. 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

A point that initially surprises some is that many of the off-diagonal terms in Eq. (5.89) are complex-valued, even when the r-space basis functions and expansion coefficients are all real. However, the momentum density is always real because each off-diagonal ij term in Eq. (5.89) is the complex conjugate of the corresponding ji term. The electron density can be written as... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

For most molecules, the small momentum expansion of the momentum density requires the full 3x3 Hessian matrix A of n( p) at p = 0. In Cartesian coordinates, this matrix has elements... 

Choosing a coordinate system that diagonalizes the Hessian matrix, the MacLaurin expansion of the three-dimensional momentum density II( p) can be written as [241]... 

For this particular two-electron example, March and Stoddart show that, in addition to the potential V(r) discussed above, which reproduces the exact Schwartz electron density p(r), it is possible to define a one-body potential which reproduces exactly the momentum density in the ground-state of the two-electron ion treated above. Naturally, because of electron correlation, this is different from V(r) above, as can be seen from its expansion for 2Zr< 1. This is given by... 

For A T < 0, Ap must contract, while for A T > 0, Ap must expand, where the contraction means an increase of low momentum density with a simultaneous decrease of high momentum density and the expansion means the density reorganization reverse to the contraction. 

In Fig. 17, these redistributions of the momentum densities are summarized for the three typical cases of diatomic interactions (Koga, 1981). In the case of attractive interactions, all the density reorganizations are initially contractions and succeedingly change into expansions in the order Ap, Ap, and Ap (Fig. 17a). For repulsive interactions with no stable molecules, the reorganizations may be expansive throughout the interactions (Fig. 17b). When there is a potential barrier (Fig. 17c), the initial... 

The definition and properties of B(r) may be summarized as follows. For simplicity, we treat the spinless one-electron wave function assuming the independent-particle model or the natural orbital expansion (Lowdin, 1955 Benesch and Smith, 1971). Based on the three-dimensional momentum density p(p),... 

In the Bom-Oppenheimer approximation, the nuclei are fixed and have zero momenta. So the momentum density IT(p) is an intrinsically one-centered function whereas p(r) is a multi-centered function. Thus one-center expansions in spherical harmonics work well for one-electron momentum densities [51-53]. The leading term of such an expansion is the spherically averaged momenmm density IIq p) defined by... 

There are two main methods for the reconstruction of 7T(p) from the directional Compton profile. In the Fourier-Hankel method [33,51], a spherical harmonic expansion of the directional Compton profile is inverted term-by-term to obtain the corresponding expansion of /T(p). In the Fourier reconstraction method [33,34], the reciprocal form factor B0) is constructed a ray at a time by Fourier transformation of the measured J(q) along that same direction. Then the electron momentum density is obtained from B( ) by using the inverse of Eq. (22). A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

Table 19.1 Coefficients in the small-p expansion, Eq. (33), of the momentum density for ground state N2...

For certain cases, it is possible to represent the time derivative of a linear variable exactly, or at least quite reasonably, in terms of a bilinear variable. No expansion or limiting process is involved the bilinear variable just happens to be well suited to express or /k. Under these circumstances, the nonlinear Langevin equation may be deduced from simple physical arguments. The cleanest examples of problems where bilinear variables arise in a fairly obvious way are diffusion problems. In any diffusion problem, self, mutual, or whatever, the linear variable of interest is a concentration, nZ [see Eq. (22)]. The time derivative of a concentration is a momentum density/mass. 

Weinstock, J. (1965). Nonanalyticity of transport coefficients and the complete density expansion of momentum correlation functions. Phys. Rev., 140, A460-A465. 

Although Ap spears at second order in the Chapman-Enskog expansion (58), it makes no contribution to the change in mass and momentum densities (55). However, Ap contributes to a first-order change in the viscons stress (78), which enters into the momentum equation at second order (70). It is therefore reasonable to construct the collision operator with the form of Ap ... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Now, in order for us to recover standard hydrodynamical behavior, we require that the momentum flux density tensor be isotropic i.e. invariant under rotations and reflections. In particular, from the above expansion we see that must be isotropic up to order... 

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space ... 

---